Derivative of the information measures was studied for the first time by Stam [1]}, where the relationship between the derivative of the differential entropy and Fisher information was presented. A connection between causal estimation error and input-output mutual information of a continuous-time additive white Gaussian noise channel, established by Duncan in [2]}. Kadota \(et\)  \(al.\)  in [3]} generalized the results reported by Duncan to the channels with feedback. Guo \(et\)  \(al.\)  in [4]} presented an explicit identity for the relationship between MMSE and the derivative of the input-output mutual information in the additive Gaussian noise channels. In [5]} the relation between estimation theory and information theory was extended to the additive non-Gaussian noise channels. In [6]}, the I-MMSE relationship was generalized to the linear vector Gaussian channel. Similar connections between the derivative of mutual information with respect to the channel parameters and the error estimation were found for the continuous-time Poisson channel in [7]} and [8]}. In [9]}, Palomar and Verdu generalized the notion of the derivative of mutual information with respect to the system parameters to the arbitrary stochastic systems, where they assumed that the system input is independent of the system parameters. Their general formula is stated in a probabilistic setting in terms of the conditional marginal input distributions given the
outputs. In [10]}, a pointwise approach to generalize the above works is considered. In [11]}, the relationship between the mutual information and the estimation error in scalar Levy channels as a more general class of observations model is expressed. The validity of I-MMSE relation for fixed finite lookahead in a continuous-time additive white Gaussian noise channel is investigated in [12]}. In [13]} an extension of the I-MMSE relation has been presented for the discrete-time Gaussian channel with feedback. An extension of the I-MMSE formula to the additive Gaussian noise MAC without feedback is given in [14]}.
